Timeline for Why did Euler consider the zeta function?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2018 at 2:54 | comment | added | LSpice | @KConrad, you can link directly to your second answer. | |
| Jun 29, 2018 at 21:33 | comment | added | KConrad | @DanPiponi, objecting to the lack of convergence at $s$ or $1-s$ is anachronistic. For someone like Euler in the 1700s, series were manipulated freely even if they make no sense to us. However, as Carlo says, Euler did use the alternating zeta-function (with terms $(-1)^{n-1}/n^s$, not $(-1)^n/n^s$, so it starts with $1$ at $n = 1$), which makes sense for $0 < s < 1$ but is strictly nonsense when ${\rm Re}(s) \leq 0$. My 2nd answer (not the accepted one) at mathoverflow.net/questions/13130/… shows how Euler found the functional equation. | |
| Jun 29, 2018 at 19:01 | comment | added | Carlo Beenakker | @Haydentech --- typo corrected, thanks. | |
| Jun 29, 2018 at 19:00 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
edited body
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| Jun 29, 2018 at 16:57 | comment | added | Haydentech | Your translation of the Latin has a small math error: you have 1/26 where it should read 1/36 | |
| Jun 29, 2018 at 16:09 | comment | added | Francois Ziegler | We should think it is key because of Euler’s product formula (also discussed in the Ayoub paper). | |
| Jun 29, 2018 at 15:49 | comment | added | Carlo Beenakker | @DanPiponi --- for that reason Euler compared $s$ and $1-s$ for the alternating zeta function $\sum_n (-1)^{n}/n^s$ | |
| Jun 29, 2018 at 15:35 | comment | added | Dan Piponi | What did Euler think it meant, to relate $\zeta$ at $s$ and $1-s$, given that at least one of these doesn't converge? | |
| Jun 29, 2018 at 15:02 | comment | added | KConrad | @Jojo there was no reason to expect the central role the generalizations of $\zeta(s)$ would play in modern number theory back in the 1700s. The first time there was any generalization, to Dirichlet $L$-functions, was in the 1830s (and then only for real $s > 1$). Zeta-functions of number fields were introduced by Dedekind in the 1870s (but were only analytically continued to $\mathbf C$ by Hecke in 1917). The central role for these constructions and more like them (for Artin representations, modular forms, etc.) was largely a 20th century realization. | |
| Jun 29, 2018 at 14:56 | history | edited | KConrad | CC BY-SA 4.0 |
changed $s-1$ to $1-s$.
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| Jun 29, 2018 at 14:49 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 29, 2018 at 8:06 | comment | added | Jojo | Thank you very much!! By the way, though we know their importance well, we should think that the Riemann zeta functions and its generalizations happen to play key roles in the modern number theory? | |
| Jun 29, 2018 at 8:00 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 29, 2018 at 8:00 | vote | accept | Jojo | ||
| Jun 29, 2018 at 7:50 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 29, 2018 at 7:45 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 29, 2018 at 7:33 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |